A classification of explosions in dimension one


 Jody Morgan
 1 years ago
 Views:
Transcription
1 Ergod. Th. & Dynam. Sys. (29), 29, doi:1.117/s c 28 Cambridge University Press Printed in the United Kingdom A classification of explosions in dimension one E. SANDER and J. A. YORKE Department of Mathematical Sciences, George Mason University, 44 University Drive, Fairfax, VA 223, USA ( IPST, University of Maryland, College Park, MD 2742, USA ( (Received 7 March 27 and accepted in revised form 26 April 28) Abstract. A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for onedimensional maps, explosions are generically the result of either tangency or saddlenode bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions. 1. Introduction For continuously varying oneparameter families of iterated maps in R n, discontinuous changes in the size of an attractor are the most easily observed type of global bifurcations, including changes in the basin boundary (metamorphosis). These changes can occur as the result of a change in stability of the recurrent set. A more general situation occurs when discontinuous changes in attractors occur as the result of a discontinuous change in the size of the recurrent set itself. Such a global bifurcation is called an explosion. For the last several years, we have been studying explosions and their properties, including a classification of explosions at heteroclinic tangencies for planar diffeomorphisms [1], and more recently a numerical study of the statistical properties of a certain kind of explosion that occurs in dimension three and higher [2]. The first major result in this paper is a full classification of which types of tangencies give rise to explosions for interval maps, a result which we think is important and new. Our research, as well as that of many others [14, 24], has been guided by a 1976 conjecture of Newhouse and Palis [19]. (See also the restatements in [1, 21].) For over thirty years, this conjecture has managed to elude proof. The conjecture says that a first bifurcation of a Morse Smale system is generically either the result of a nonhyperbolic periodic orbit, or the result of a tangency between stable and unstable manifolds of fixed or periodic orbits. We make the following reformulation of this conjecture for
2 716 E. Sander and J. A. Yorke global bifurcations: for generic planar diffeomorphisms, all explosions occur through the following two local bifurcations: saddlenode bifurcations and tangency bifurcations. The second major result of this paper is that the the analogous conjecture is true for generic smooth interval maps. This is in contrast with a recent result of Horita et al [14], showing that in a probabilistic sense, the Newhouse Palis conjecture is not true for circle maps. In a broader sense, we are hopeful that the insight gained from the onedimensional case will give rise to insights leading to a better understanding of explosions in two dimensions. It is clear that an isolated saddlenode bifurcation for either a fixed point or periodic orbit gives rise to a local explosion, since new periodic points appear. However, in many cases in both one and higher dimensions, a saddlenode bifurcation also gives rise to a global bifurcation. The set of recurrent points changes discontinuously as the parameter is varied at points not contained in the saddlenode periodic orbit. For example, the period three window in the chaotic attractor for the logistic map corresponds to an explosion: for parameter values less than a bifurcation value, there is a global attractor which comprises the full recurrent set. It consists of an interval. After the bifurcation parameter, the global attractor consists only of a period three orbit, and the full recurrent set is a nowhere dense Cantor set within the attractor interval. An explosion occurs at the bifurcation parameter, and it is due to a saddlenode bifurcation at a point inside the global attractor. Saddlenode bifurcations on invariant circles are another wellstudied example of this phenomenon. In all dimensions, explosions due to a homoclinic or a heteroclinic bifurcation essentially occur due to the creation of a homoclinic tangle when stable and unstable manifolds intersect transversally. In one and two dimensions, the transition between no intersection and transverse intersections involve tangencies between the stable and unstable manifolds of fixed or periodic points (cf. [1]). For example, the Hénon map and Ikeda map contain wellstudied examples of explosions which are a result of homoclinic bifurcations. In higher dimensions, such bifurcations can occur without tangencies [13]. Such a bifurcation leads to unstable dimension variability [2]. In our previous work for planar maps, we gave a precise classification for which types of tangencies for heteroclinic cycles will result in explosions. We called this class of cycles crossing cycles, because the different stable and unstable manifolds involved in the cycle lie across the tangency from each other. We show here that the same results hold for interval maps. Our main results are as follows. THEOREM 1. (Explosions at tangencies) For generic oneparameter families of smooth maps of the interval with homoclinic or heteroclinic tangencies (hypotheses H1 H6), explosions occur if and only if there is an isolated crossing orbit. THEOREM 2. (General explosion classification) Explosions within generic oneparameter families of smooth maps of the interval (hypotheses H1 H3) are the result of either a tangency between stable and unstable manifolds of fixed or periodic points or a saddlenode bifurcation of a fixed or periodic point. The paper proceeds as follows: in 2, we give basic definitions of explosions and homoclinic tangencies. We are considering the particular recurrence class of chain recurrent points, defined in this section. A motivation for the choice of setting is given in
3 A classification of explosions in dimension one 717 the section as well. In 3, we prove the explosions at tangencies theorem. In 4, we prove the General explosion classification theorem. Our results rely on a very sophisticated and welldeveloped theory for the dynamics of interval maps. We briefly state the necessary background and results in the course of the proof. 2. Basic definitions We now give some formal definitions of concepts described in the introduction. Let f : (I J) (R R) R be a smooth oneparameter family of maps. We exchangeably write two types of notation: f (x, λ) = f λ (x). For the definition of an explosion it is more natural to use the concept of chain recurrence rather than recurrence. The relationship between chain recurrence and other types of recurrence is discussed in Remark 1 and in [1]. Definition 1. For an iterated function g, there is an ɛchain from x to y when there is a finite sequence (z, z 1,..., z N ) such that z = x, z N = y, and d(g(z n 1 ), z n ) < ɛ for all n. If there is an ɛchain from x to itself for every ɛ > (where N > ), then x is said to be chain recurrent [11, 12]. The chainrecurrent set is the set of all chainrecurrent points. For a oneparameter family f λ, we say (x, λ) is chain recurrent if x is chain recurrent for f λ. If for every ɛ > there is an ɛchain from x to y and an ɛchain from y to x, then x and y are said to be in the same chain component of the chainrecurrent set. The chainrecurrent set and the chain components are invariant under forward iteration. We now define an explosion bifurcation in the chainrecurrent set. Such a definition can be formulated for the nonwandering set as well. Definition 2. (Chain explosions) A chainexplosion point (x, λ ) is a point such that x is chain recurrent for f λ, but there is a neighborhood N of x such that on one side of λ (i.e. either for all λ < λ or for all λ > λ ) no point in N is chain recurrent for f λ. (All explosion points in this paper are chain explosions, so we sometimes drop the qualifier chain.) Remark 1. (Bifurcations and explosions) Explosions have been used in the past to study global bifurcations for planar diffeomorphisms [1, 2 23]. They are closely related to the loss of structural stability, since interesting dynamics only occur where there is recurrence. If f λ is a structurally stable function in a oneparameter family on a compact set, then for any x, (x, λ ) is not an explosion point. For smooth maps on compact domains, the chainrecurrent set is upper semicontinuous, whereas the closure of the set of hyperbolic periodic points is lower semicontinuous. If the sets are equal, then they change continuously [1]. In studying global bifurcations, the setting of explosions has an advantage over considering structurally stable systems in that we are not restricted to considering first bifurcations. Explosions can occur for example at the limit point of a sequence of bifurcations. However, there are bifurcations which are not explosions. For example, a point of period doubling is not an explosion point, although it is a bifurcation. In this paper, we do not discuss local bifurcations.
4 718 E. Sander and J. A. Yorke Our work considers chain explosions, whereas others have considered explosions in the nonwandering set. These concepts are quite closely related: the set of periodic points is contained in the recurrent set, which is contained in the nonwandering set. The nonwandering set is contained in the chainrecurrent set. There are examples showing that all the containments are strict. However, for a generic smooth diffeomorphism on a compact manifold, the chainrecurrent set and the nonwandering set are both equal to the closure of the set of hyperbolic periodic orbits (cf. [1, 1.2]). Our choice of chain explosions as opposed to nonwandering explosions is motivated by the following useful recharacterization of the property of upper semicontinuity: if (x, λ) is a chainexplosion point, then x is chain recurrent for parameter λ. This is not true for the nonwandering set or for the closure of the hyperbolic periodic points. Using this property, in our technical arguments we are able to separate out the dynamics and the parameter change, allowing us to confine many of our arguments to studying the chainrecurrent set at the parameter for which the chain explosion occurs. Our investigation is practically equivalent to studying maps (without parameters) for which the chainrecurrent set is strictly bigger than the recurrent set, provided that the map is sufficiently nice that it could be seen in generic oneparameter families. Remark 2. In order to show that (y, λ) is not a chainexplosion point, it is sufficient to show that y is in the closure of the hyperbolic periodic orbits for f λ. Remark 3. In the above definition, at f λ, x is not necessarily an isolated point of the chainrecurrent set. For example, at a saddlenode bifurcation on an invariant circle, the chainrecurrent set consists of two fixed points prior to bifurcation and the whole circle at and in many cases after bifurcation. The chainrecurrent set is not invariant under backwards iteration of a noninvertible map. Thus explosion points are not preserved under iteration, forward or backward. The following remark states what is guaranteed by the fact that chain recurrence is preserved under forward iteration. Remark 4. Let (x, λ ) be a chainexplosion point for f. Specifically, there exists δ > such that there is no chainrecurrent point in B δ (x) for all λ < λ, but x is chain recurrent at λ. Then f (x) is chain recurrent at λ, but f (x) may also be chain recurrent for λ < λ. In contrast, if x 1 is a preimage of x, then there is a δ 1 > such that no point in B δ 1 (x 1 ) is chain recurrent for all λ < λ. Note that x 1 may not be chain recurrent at λ. We now give definitions of homoclinic and heteroclinic points. Note that for a diffeomorphism, homoclinic and heteroclinic orbits require the existence of saddle points with stable and unstable manifolds of dimension at least one. Thus they can only occur in dimension two or greater. However, for noninvertible maps, it is possible to have a fixed or periodic point with a onedimensional unstable manifold and a zerodimensional stable manifold. Marotto terms such points snapback repellers [17]. It is not possible to reverse these stable and unstable manifold dimensions; the existence of a homoclinic orbit to an attracting fixed point requires a multivalued map [25]. The following definition of homoclinic points for interval maps is depicted in Figure 1.
5 A classification of explosions in dimension one 719 x y FIGURE 1. A onedimensional map with a repelling fixed point x where f (x ) >. The point y is a homoclinic point, since f k (y) = x (for k = 1), and there exists a sequence of successive preimages of y converging to x. Definition 3. (Homoclinic points) Let f : R R be a smooth function with a repelling fixed point x. Let y be a point in the unstable manifold of x and an integer K > such that f K (y) = x. Then y is a homoclinic point to x. If x is periodic with least period m, then the same definition applies by replacing f with f m. Remark 5. If p is a hyperbolic fixed point, and the forward limit set of a point z in the unstable manifold of p includes p, then there are two possibilities: (1) a finite iterate of z is equal to p, (2) the limit set of z contains points other than p. In this second case, z is not referred to as a homoclinic point, since its limit set is larger than just p. This case is considered in later sections of this paper. For diffeomorphisms, all orbits through homoclinic points are homoclinic orbits. For onedimensional maps, there may be many nonhomoclinic orbits through a homoclinic point. Definition 4. (Homoclinic orbits) Let f, x, and y be as in the above definition of a homoclinic point. An orbit (z k ) k= is a homoclinic orbit through y if the following conditions are satisfied: z = x, z K = y for some K, and for all k N, f (z k ) = z k+1, and lim k z k = x. Since the stable manifold of a homoclinic point is zerodimensional, a homoclinic tangency is a tangency of the graph of the map at a homoclinic point. Homoclinic tangencies are depicted in Figures 2 and 3. Definition 5. (Homoclinic tangencies) Let f, x, y, and (z k ) k= be as in the above definition of a homoclinic orbit. The point w = z L is a homoclinic tangency point if the graph of f is tangent to the horizontal line at w. 3. Explosions at homoclinic tangencies This section classifies explosions occurring via homoclinic tangencies. The results are stated for fixed points, but the same results hold for homoclinic orbits for periodic points
6 72 E. Sander and J. A. Yorke x U right w FIGURE 2. A onedimensional map with a repelling fixed point x with f (x ) >. The point w is a homoclinic tangency point. For the particular tangency depicted here, w is contained in a noncrossing orbit (Definition 7), and thus by Theorem 5 w is not an explosion point. x U right y w FIGURE 3. In a homoclinic orbit as in Figure 2, the tangency point w does not need to be the immediate preimage of the fixed point. Here, w is the second preimage of x. with least period m if f is replaced by g = f m. Throughout this section, we make the following hypotheses. The first hypothesis is a smoothness assumption. The second and third hypotheses are generic assumptions for oneparameter families. The fourth is a notational convention for the existence of a homoclinic orbit. The fifth hypothesis is generic for oneparameter families containing a homoclinic orbit. H1 f : (I J) (R R) I is a C 1 smooth family of C 2 interval maps. For a fixed parameter λ, denote f := f λ. H2 Assume H1. Assume that there are no intervals on which f is constant. Assume also that critical points of f are nonflat. H3 Assume H1. Assume that for f, there is at most one of the following. (a) A nonhyperbolic fixed point or periodic orbit. We assume generic behavior as a parameter is varied. That is, any nonhyperbolic periodic point is a codimension one perioddoubling or saddlenode bifurcation.
7 A classification of explosions in dimension one 721 (b) One critical point which comprises a tangency between stable and unstable manifolds of fixed points or periodic orbits. We assume generic behavior as a parameter is varied. That is, with variation of parameter, the critical point moves from above to below (or below to above) the placement of the periodic point. H4 Assume H1, and that for each λ, x λ is a repelling fixed point for f λ. Denote x := x λ. Assume that for f, y is homoclinic to x. H5 Assume H1 and H4. At λ = λ, the homoclinic orbit containing y contains only one critical point. The following theorem says that if at a repelling periodic point a map has negative derivative then no homoclinic points are explosion points. THEOREM 3. (No explosions for orbits with negative derivatives) Assume that f λ is a family of maps satisfying H1 H5, and f λ (x ) <, then (y, λ ) is not an explosion point. Proof. We show that y is contained in the closure of the hyperbolic periodic points of f λ. If y is contained in a homoclinic orbit without a tangency, then the homoclinic orbit is preserved under perturbation, which automatically implies that (y, λ ) is not an explosion point. Thus we assume that y is contained in a homoclinic orbit containing a homoclinic tangency point w. As before, denote this orbit by (z k ) k=, where z = x, lim k z k = x, and K and L are such that y = z K and w = z L. Fix a neighborhood U of y. By H5, there exists a sequence of neighborhoods U k z k, such that: (i) U K U; (ii) for k L, f (U k+1 ) = U k and the map is injective; and (iii) for k < L, U L k = f k (U L). Let V 1 = f L (U L \ w) = U \ x. By H2, V 1 is an interval on one side of x. By assumption f (x ) <, so if the {U k } are sufficiently small, V 2 = f L+1 (U L \ w) is an interval on the other side of x. That is, V 1 V 2 {x } is a neighborhood of x. For some sufficiently large J, U J U. In addition, f J (U J ) = U. Thus for k J, U k contains a periodic point. Thus there is a periodic point in U. Since U was arbitrary, there are periodic points p j with period n j such that lim j p j = y and lim j n j =. By H3, for j sufficiently large, the periodic points are hyperbolic. Therefore (y, λ ) is not an explosion point. Now consider the positive derivative case. Let w be a homoclinic tangency point contained in homoclinic orbit (z k ) k=. Since the eigenvalue of x is positive, there is an M sufficiently large that (z k ) k=m lies entirely on one side of x. That is, the homoclinic orbit converges to the fixed point along one branch of the unstable manifold. We denote this by saying that the homoclinic orbit is contained in the local right or left branch of x, as formalized in the following definition. Definition 6. (Two branches of the unstable manifold) Let x be a repelling fixed point for f C 2, with f (x ) >. The local left and right branches of the unstable manifold of x are disjoint. Define U left and U right to be the respective unions of images of local left and right manifold branches.
8 722 E. Sander and J. A. Yorke w U left x U right i j FIGURE 4. The tangency point w is contained in U left U right, and is thus not an explosion point. However, the preimages i and j of w are explosion points. Remark 6. The union of U left and U right is the entire unstable manifold of x. By the intermediate value theorem U left and U right are intervals. If (U left U right ) \ x is not empty, then the intersection must contain either U left or U right. For example, U right may contain points both to the left and to the right of U left. See Figures 4 and 5. From the proof of Theorem 3, it is clear that to study chain explosions in homoclinic orbits, it is sufficient to consider homoclinic orbits containing tangency points. We formalize the notation in the following hypothesis: H6 For a family satisfying H1 and H4, at f, point w is a homoclinic tangency point to x contained in (at least one) homoclinic orbit (z k ) k=. Let L be such that w = z L. LEMMA 1. (No explosions when manifold branches intersect) Assume H1 H6. Assume that f (x ) > and that w U right. If for any neighborhood N w, the sets f L (N) and the local right branch of x contain points in common, then (w, λ ) is not an explosion point. Proof. The details of this proof are similar to the negative derivative case. Preimages of any small neighborhood N of w are contracting and in the local righthand manifold branch. Thus the Lth image of N includes a shrunk preimage of N. Therefore N contains a periodic point, which by H3 and H5 is hyperbolic. Since N is arbitrary, (w, λ ) is not an explosion point. Remark 7. The theorem above is also true when U right is replaced by U left. THEOREM 4. Assume H1 H6, and that w is contained in U right U left. Then (w, λ ) is not an explosion point.
9 A classification of explosions in dimension one 723 z U left w x U right FIGURE 5. In this figure, w is a homoclinic tangency point contained in a crossing orbit. Thus w lies in U right, but not U left. By Theorem 6, w is an explosion point. As depicted here, the preimage z of w is also an explosion point. Proof. Take a small neighborhood N of w. (N) is either a local left or a local right branch of x. Since w is contained in both U left and U right, there is a shrunk preimage of N contained in f L(N). Therefore by Lemma 1, (w, λ ) is not an explosion point. As mentioned in the introduction, in our previous work we gave a useful geometric method of approaching chain explosions in homoclinic orbits in two and three dimensions, termed crossing cycles. In two dimensions, we showed that a crossing cycle is necessary and sufficient for a chain explosion to occur [1]. The analogous statements are true in one dimension, as in the theorem below. Crossing and noncrossing orbits are shown in Figures 5 and 2 respectively. Definition 7. (Crossing orbits) Assume H1 H6. We call the homoclinic orbit (z k ) a crossing orbit if for sufficiently large k, z k U left \ U right (respectively U right \ U left ), and for any interval M which is a local right (respectively left) branch of the unstable manifold to x, there is a sufficiently small neighborhood N of the tangency point z L such that f L (N) is contained in M. A homoclinic orbit that is not crossing is called a noncrossing orbit. Remark 8. Note that there is a geometric interpretation for a crossing orbit: if a crossing orbit has a tangency point at w = z 1, the definition is equivalent to saying that z k U left \ U right (respectively U right \ U left ), and f is locally above (respectively below) the horizontal line at z 1. If w = z L for L > 1, then for g = f L 1 there is a tangency point at z 1, and one can draw the same geometric conclusion for g at z 1. THEOREM 5. (Explosions imply crossing orbits) If H1 H6 hold and (w, λ ) is an explosion point, then (z k ) k= is a crossing orbit. f L
10 724 E. Sander and J. A. Yorke Proof. If at λ = λ there is a noncrossing homoclinic orbit containing w, then under f L the image of a small neighborhood of w contains a shrunk preimage of itself. The result now follows from Lemma 1. The results stated so far give necessary conditions for an explosion point. We now give a converse to these. The following theorem gives sufficient conditions for an explosion. THEOREM 6. (Crossing bifurcations and explosions) Assume H1 H6. If every homoclinic orbit containing w is a crossing orbit, then (w, λ ) is an explosion point. Proof. Assume H1 H6 and that every orbit containing the homoclinic tangency point w = z L is a crossing orbit. The point w is a homoclinic point for f, so w is chain recurrent for λ = λ. Any sufficiently small neighborhood N of w is such that f L (N) is contained in the local left (respectively right) branch of x. If w U left U right, then there exist a set of preimages q k for k > L of w converging to x from the left (respectively right) such that the union of (q k ) for k > L and (z k ) for k L forms a noncrossing orbit, contradicting our hypothesis. Therefore either w U right \ U left or w U left \ U right. For specificity, let w U right \ U left. Since U right is an interval, any neighborhood N of w contains points in U right other than w. Choose N sufficiently small such that under f L, the image of N is contained only in the local left branch of x. Note that U right is forward invariant. Thus there are points in f L(N) contained in U left U right \ {x }. By Remark 6, since U left U right contains points other than x, either U left U right or U right U left. Since w is contained in U right but not in U left, U left U right, and the inclusion is strict. Thus the right endpoint of U left is x. Furthermore, no points in the interior of U left map to x under f, since that would imply a second simultaneous homoclinic tangency, violating H3b. Now consider the left endpoint x L of U left. Since U left is forward invariant, f (x L ) U left. This leaves two possibilities. (i) x L maps into the interior of U left. If this is the case, there is a neighborhood of x L which also maps into U left. (ii) x L is a fixed point. In this second case, since there is already a homoclinic tangency at w, H3 implies that x L is a hyperbolic fixed point. We assume x L is a repelling fixed point (to get a contradiction). Specifically, we will show in the next paragraph that in this case, there is a critical point in U left mapping onto x L. That means there is a heteroclinic tangency between x and x L. This is a contradiction, since w was assumed (H3) to be the only tangency point between stable and unstable manifolds. Therefore, we can conclude that if x L is a fixed point, then it is an attracting fixed point. Here is the proof that if x L a repelling fixed point then there is a critical point in U left mapping to x L : if a point r in the interior of U left maps onto x L, then r is a critical point. Specifically, since U left is forward invariant, a neighborhood of r maps to the local right branch of x L. If no such r exists, then any interval of the form (x a, x ) contains a sequence of points c k such that lim k f k(c k) = x L. The limit approaches x L from the right since x L is the left endpoint of a forwardinvariant set. Since x L is assumed to be a repelling fixed point, we know that there is an interval (x L, x L + b)
11 A classification of explosions in dimension one 725 which is monotone increasing under f. This implies that for k sufficiently large, there is a minimum j k 1 such that f j k+1 (c k ) > f j k (c k), and f j k (c k) f k(c k). The latter condition implies that lim k f j k (c k) = x L. Note that since j k is the minimum such value, f j k 1 (c k ) > f j k (c k). Therefore f j k 1 (c k ) is not in (x L, x L + b). There is a convergent subsequence of f j k 1 (c k ). Call the limit point q. Then f (q) = x L, but q x L. Thus there is a critical point in the interior of U left mapping to x L. (iii) There appears to be a third possibility for x L : f (x ) = x. In the setting of this theorem, this third possibility cannot occur by the following argument: as in (ii), if a point r in the interior of U left is such that f (r) = x L, then r is a critical point. Since x L is a homoclinic point, this is ruled out since it would be a simultaneous homoclinic tangency. Assume there is no such r. Since x L is an endpoint of U left, there is a sequence of points c k in the interior of U left such that f (c k ) converges to x L. A convergent subsequence of c k exists, and has a limit point r such that f (r) = x L. Unlike in (ii), x L is not a fixed point, so no fancy argument is needed to show that r x L (and r x ). Therefore r is in the interior of U left, which contradicts our assumption. This implies that (iii) cannot happen. So far we have shown that either (i) there is an infimum value of x L in U left, and x L maps into the interior of U L or (ii) x L is an attracting fixed point. In either case, for λ close to λ, there is a continuation of the left endpoint of U left, which we denote xl λ. In case (i), there is a continuation of the local infimum value xi λ. In case (ii), there is a continuation of the fixed point which we denote also by xi λ (since the two cases do not appear at the same time). The interior of Uleft λ for each λ is contained in the interval [xλ I, xλ ]. This is clear in case (i). In case (ii), x L is attracting, and at λ there are no critical points mapping to x L from inside of U left. Thus small perturbations of the map have the same property. Therefore for small ɛ >, no ɛchains leave the interval [xi λ, xλ ] to the left. In fact, since x is repelling with no tangency points in U left, there exists a δ > such that for any sufficiently close λ, for all δ < δ, there is an ɛ > such that no ɛchains leave [xi λ, xλ δ]. We have assumed that w is a generic and unique homoclinic tangency point for f. Therefore for λ near λ prior to tangency, f L (N) is contained in the interior of the local left branch and not in the local right branch of x. Thus for some δ < δ, fλ L (N) is contained in [xi λ, xλ δ]. This implies that there is an ɛ > such that no ɛchains carry points from N to itself. Thus (w, λ ) is an explosion point. We are interested not only in explosion points which are themselves tangency points, but also in explosion points far from tangencies which are caused by tangencies. Theorem 3 contained such a result, but the other results were specifically about the tangency points. It is now straightforward to combine the previous results with Remark 4 to get results for general homoclinic points. Since the chainrecurrent set is invariant under forward iteration, the image of a noncrossing tangency point is a nonexplosion point. However, there may be explosion points with iterates that are nonexplosion points. For example, Figure 4 shows points of explosion which are not tangency points but are preimages of tangency points. THEOREM 7. (Explosions when manifold branches do not intersect) If H1 H6 hold, and there exists a k > such that (z k, λ ) is an explosion point, then the orbit (z k ) is a crossing orbit.
12 726 E. Sander and J. A. Yorke Proof. If the orbit is a noncrossing orbit, then there are hyperbolic periodic orbits limiting on every point in the orbit. THEOREM 8. If H1 H6 hold, and (w, λ ) is an explosion point, then any preimage z k in the homoclinic orbit of w is such that (z k, λ ) is an explosion point. Proof. If (w, λ ) is an explosion point, then by Lemma 4, z k is not chain recurrent for λ < λ. At λ, z k is contained in a homoclinic orbit, and thus chain recurrent. Remark 9. This theorem only mentions preimages of w which are contained in a homoclinic orbit at λ. There may be other preimages of w which are not contained in any homoclinic orbit, and are thus not chain recurrent at λ. Remark 1. Assume H1 H6, and that at λ the tangency point w is contained exclusively in one manifold branch of x. Then at λ = λ, either tangency point w U right, and U left contains no tangencies to fixed points or periodic orbits, or tangency point w U left, and U right contains no tangencies to fixed points or periodic orbits. This follows from the fact that w is only contained in one of U left and U right, so by H3 there is no tangency in the other manifold branch. THEOREM 9. Assume H1 H6. Assume that for λ = λ, (z k ) k= is a crossing orbit, but w is also contained in some other homoclinic orbit (α k ) k= which is a noncrossing orbit. Thus (w, λ ) is not an explosion point. Then the following statements hold: (i) for all m >, ( f m (w), λ ) is not an explosion point; (ii) if there exists J > L such that z J is contained neither in (α k ) k= nor in any other noncrossing orbit, then for all k > J, (z k, λ ) is an explosion point. Proof. The first statement holds since at λ, any image of w is contained in (α k ) k=, which is a noncrossing orbit. Thus by Theorem 7, this image is not an explosion point. Let J > L be as in Part (ii) of the theorem. For specificity, assume that z J U right. As in the proof of Theorem 6, since z J is not contained in any noncrossing orbit, it is not contained in U left. A neighborhood of z J maps into U left under f J. If the tangency point w is in U right \ U left, then the rest of this proof is exactly the same as the proof of Theorem 6, and we conclude that (z J, λ ) is an explosion point. If w U left U right, then the penultimate forward orbit of w: (x k ), 1 k L is contained in U left U right \ {x }. Therefore, the options for x L include the two options listed in the proof of Theorem 6, in addition to the third option that x L is included in the forward orbit of w. In this case, for λ prior to tangency, the left endpoint of U left moves such that it maps into the interior of U left. Let xi λ denote the minimum preimage of the continuation of the critical point of w in this third case. Whether or not x L is in in the forward orbit of w, for all λ sufficiently near λ prior to tangency, there exists a δ λ > such that for all δ < δ λ, there is an ɛ > such that no ɛchain leaves [xi λ, xλ δ]. The rest of the proof that z J is an explosion point is the same as in Theorem 6. For all k > J, z k has all the same key properties as z J, implying that (z k, λ ) is an explosion point. This completes the classification of when homoclinic points are explosion points.
13 A classification of explosions in dimension one 727 Unlike the planar case, in one dimension the heteroclinic case reduces to the homoclinic case. That is, if there is a transverse heteroclinic cycle including an orbit from {p} to {q}, which are hyperbolic periodic orbits, then both periodic orbits must be repellers. Further, the unstable manifold of {p} contains the unstable manifold of {q}. Since we assume only one tangency, a heteroclinic tangency point is also a homoclinic tangency point. 4. General explosion classification There are many previous results on the structure of ωlimit sets and chainrecurrent sets for interval maps. Block and Coppel [5] showed that the chainrecurrent set for maps of the interval can be classified as the set of points {x x Q(x)}, where Q(x) is the intersection of all asymptotically stable sets containing the limit set ω(x). They showed that Q(x) is either an asymptotically stable periodic orbit, a set of asymptotically stable iterated intervals, or a special type of set known as a solenoidal set. However, this classification is not as useful as it would appear, since Q(x) is not the set of points in the chain component containing x. Block [4] also proved that an interval map has a homoclinic point if and only if it has a periodic point with period not a power of two. Block and Hart [6] improved on this result to show the existence of a homoclinic point to a given power of two implies a cascade of homoclinic bifurcations. Further, if a family of maps changes from zero to positive entropy, then there is a cascade of homoclinic bifurcations. Of most relevance to the topic of this paper are the works of Sarkovskii, Mañé [16], Blokh [7, 9], and Blokh, Bruckner, Humke, and Smítal [8], where the detailed structure of all possible ωlimit sets is studied. The ωlimit set can be a Cantor set known as a basic set (Definition 9). Another interesting case occurs when the ωlimit set of a point is a limit of a perioddoubling cascade, known as a solenoid (Definition 1). See [3, 15] for a detailed characterization of solenoids. Blokh [9] showed that for C 2 smooth maps, ωlimit sets are either periodic orbits, periodic transitive intervals, subsets of basic sets, or solenoids. We use this result to systematically show that for all possible explosions, there are saddlenode or tangency bifurcations. The following definition describes points at which jumps in an ɛchain are required. We call these points barricades, as they serve to obstruct orbits. For example, at the bifurcation parameter, a saddlenode point blocks the points on one side from reaching the other side. Definition 8. (Barricade) Assume H1 and H2. Let z be any point. Let S ɛ = ω(b ɛ (z)). A point y lim ɛ S ɛ is called a barricade for z if it is blocking the orbit of z. That is, let Z ɛ = ω(b ɛ (y)). Then lim ɛ Z ɛ contains points not contained in lim ɛ S ɛ. We consider a point such that the ωlimit set is a fixed point or periodic orbit. THEOREM 1. Assume H1 H3. Assume that for f, z is a point such that ω(z) is equal to a fixed point or periodic orbit {p}, and {p} is a barricade. Then {p} is either nonhyperbolic or is the image of a critical point. Proof. Assume that p is a fixed point, since otherwise we can let g = f n. Since p is a barricade, p S ɛ, and lim ɛ Z ɛ lim ɛ S ɛ (using the notation from Definition 8). This implies that p cannot be an attracting fixed point, since for an attracting fixed point lim ɛ Z ɛ = p. Thus if p is hyperbolic, it must be repelling, meaning that z is a preimage
14 728 E. Sander and J. A. Yorke of p. Define K by f K (z) = p. Since p is a barricade, for small ɛ, f K (B ɛ (z)) is an interval on one side of p, implying that p is in the orbit of a critical point. The previous theorem only indicates that a critical point exists. It still remains to be shown that the critical point is actually a homoclinic or heteroclinic point. We use the fact that all ωlimit sets for interval maps have been classified. The following theorem is useful in the proof of several subsequent theorems. THEOREM 11. Assume H1 H3, and that M is an invariant interval under f, such that for all ɛ >, there is an ɛchain from a point x M to a point y / M. Then there is an ɛchain from an endpoint e of M to y, where e is fixed or period two. Furthermore, if x is not an endpoint of M, then either e is nonhyperbolic; or e is repelling, and there is an orbit of a critical point in M mapping onto e. Proof. Since f (M) = M, the only way for an ɛchain to exit M is through an ɛjump across one of the endpoints. Call the endpoint e. Thus there is a chain from e to y. Assume e does not map to itself or to the other endpoint of M. Then f (e) is contained in the interior of M. But this means that no small ɛjump at e exits M, which is a contradiction. Thus e can be chosen to be either a fixed point or a period two point. The orbit of e cannot be attracting, since then there would not be ɛchains from e to any other point. If the orbit of e is hyperbolic, then it is repelling, and there is a chain from x to e. If the orbit of x includes e, and x is not an endpoint of M, then the orbit of x contains a critical point, since M is invariant. If the orbit of x does not include e, then the limit set of x contains more than just a repelling periodic orbit. The following result shows what happens if the backward limit set of a point contains a periodic orbit. THEOREM 12. Assume H1 H3. Let {p} be a fixed point or periodic orbit for f which is hyperbolic. Let z be a point such that for all ɛ > there is an ɛchain from {p} to z, but z is not in the unstable manifold of {p}. Then there is a homoclinic tangency to a periodic orbit. Proof. Assume without loss of generality that p is a fixed point (since otherwise, we can use the same proof for an iterate of f ). Since p is hyperbolic, it is repelling. The unstable manifold of p is an invariant interval, denoted U(p). Since z / U(p), there is a barricade point for p. By Theorem 11, there must be a barricade point which is a periodic endpoint of U(p), with a critical point in U(p) mapping to the endpoint. That is, there is a homoclinic tangency point for the periodic endpoint. Combining Theorems 1 and 12, we conclude that if an explosion occurs at a point (z, λ ) such that ω(z) is equal to a periodic orbit, then there is either a saddlenode bifurcation point or a tangency between stable and unstable manifolds. We now consider more general ωlimit sets. First consider the case where the ωlimit set of a point is an interval. THEOREM 13. Assume H1 H3, for f, z is such that ω(z) is an interval M, and that there exists x M such that e is a barricade for z. Then e is an endpoint of the interval, e is fixed or period two, and if e is hyperbolic, then there is a homoclinic tangency to e.
15 A classification of explosions in dimension one 729 Proof. Since ω(z) = M, f is transitive on M. Therefore the unstable manifold of any repelling periodic orbit in M contains all of M. A barricade must not be in the interior of ω(z). Thus e is an endpoint of M. The result now follows from Theorem 11. We now consider the case of an ωlimit set that is contained in an invariant interval but is nowhere dense. Definition 9. (Basic set [7]) Assume H1. Let M = n k=1 I k be an nperiodic cycle of intervals for a function f. Define B(M, f ) = {x M for every relative neighborhood U of x in M, orb(u) = M}. If B(M, f ) is infinite, then it is called a basic set. THEOREM 14. Assume H1, H2, and H3a. Assume that (z, λ ) is an explosion point, and ω(z) is nowhere dense and is contained in a basic set B(M, f ). Then ω(z) is a periodic orbit. Proof. Assume that (z, λ ) is an explosion point, and that ω(z) B(M, f ). Assume that z is contained in an interval complementary to B(M, f ). Blokh [7] considers the class of interval maps such that if I is a wandering interval, then ω(i ) is a periodic orbit. This class of maps includes smooth interval maps [7] with nonflat critical points [18]. Under the assumption that ω(z) is a nowhere dense set contained in a basic set B(M, f ), [7, Property C] shows that if a point z is not contained in B(M, f ) but ω(z) is, then ω(z) is a periodic orbit for f. Assume ω(z) is nonperiodic, meaning z is contained in the basic set. By [9], B(M, f ) is contained in the closure of the periodic orbits for f. Using H3a, there is a sequence of hyperbolic periodic points converging to z. Thus (z, λ ) is not an explosion point. By the above theorem combined with Theorems 1 and 12, if an explosion point has an ωlimit set which is a basic set, then there is either a saddlenode point or a tangency. The last possibility for an ωlimit set is a solenoid, as in the following definition. Definition 1. (Solenoid) Assume H1. Let M j = n j k=1 I j k be a nested sequence of cycles of intervals for a function f with least period n j, lim j n j =. Thus f n j (I j 1 ) = I j 1, n j is increasing, and for each j, I j+1 1 I j 1. If the set S = j=1 M j is nowhere dense, then S is called a solenoid or Feigenbaumlike set. Jiménez López has shown that solenoids are the boundary of chaos and order [15]. Blokh [9] demonstrated that solenoids and basic sets are disjoint. We prove the following result. THEOREM 15. Assume H1, H2, and H3a. Assume that (z, λ ) is an explosion point, and ω(z) is a solenoid S. Then there is an infinite sequence of periodic orbits which are barricades with associated tangencies. Proof. Since solenoids and basic sets are closed, invariant, and disjoint, there is a neighborhood of solenoid S containing no basic sets. By definition, there is an infinite nested sequence of invariant cycles of intervals M j containing S. As before, for interval maps such that a nonwandering interval I has ω(i ) periodic, Blokh [7] proved that the periodic orbits are dense in a neighborhood of S, meaning that z is not contained in S. Thus
16 73 E. Sander and J. A. Yorke for all j sufficiently large, for all ɛ > there is an ɛchain from points in M j to z, but z is not contained in M j. By Theorem 11, there exists e j, an endpoint of an interval the cycle of M j which is periodic. By hypothesis H3a, for sufficiently large j, e j is hyperbolic. Since for all ɛ > there is an ɛchain from e j to z, e j is a repeller for large j. There is an orbit of a critical point in M j mapping onto e j. Furthermore, z is not contained in the unstable manifold of e j, since there is a nested sequence of invariant M j not containing z. Theorem 12 implies that the critical point to e j is a point of homoclinic tangency. Remark 11. If f has a finite number of homoclinic and heteroclinic tangencies, as assumed in H3b, then the above theorem shows that there are no forward chains from solenoid S to a point outside of S. Remark 12. We have shown that there is either a tangency or a nonhyperbolic critical point contained in the same chain component as z. Under a generic hypothesis (H3a), a nonhyperbolic periodic orbit is either codimensionone saddlenode or perioddoubling bifurcation. In fact, such an orbit is not a perioddoubling point, since the periodic orbit at a perioddoubling bifurcation point is attracting. We now combine the results of this section to give a proof of the General explosion classification theorem. Proof of the General explosion classification theorem. Assume that (z, λ ) is an explosion point. The only possibilities for ω(z) are a periodic orbit, a cycle of intervals, a nowheredense basic set, and a solenoid. Above, we have shown that in any of these cases, there is a periodic barricade point for z which is either nonhyperbolic, or there is a homoclinic or heteroclinic tangency. In fact, the case of a solenoid is ruled out by H3b. By Remark 12, a nonhyperbolic periodic orbit must necessarily be a saddlenode point. This completes the proof the theorem. Acknowledgements. ES would like to thank Alexander Blokh and Victor Jiménez López for their helpful suggestions. JAY was partially supported by NSF Grant DMS ES was partially supported by NIHNIMH Grant MH7952. REFERENCES [1] K. Alligood, E. Sander and J. Yorke. Explosions: global bifurcations at heteroclinic tangencies. Ergod. Th. & Dynam. Sys. 22(4) (22), [2] K. Alligood, E. Sander and J. Yorke. Threedimensional crisis: crossing bifurcations and unstable dimension variability. Phys. Rev. Lett. 96(24413) (26). [3] L. Alsedà, V. J. López and L. Snoha. All solenoids of piecewise smooth maps are period doubling. Fund. Math. 157(2 3) (1998), (Dedicated to the memory of Wiesław Szlenk.) [4] L. Block. Homoclinic points of mappings of the interval. Proc. Amer. Math. Soc. 72(3) (1978), [5] L. Block and W. Coppel. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513). Springer, Berlin, [6] L. Block and D. Hart. The bifurcation of homoclinic orbits of maps of the interval. Ergod. Th. & Dynam. Sys. 2(2) (1982),
17 A classification of explosions in dimension one 731 [7] A. Blokh. Density of periodic orbits in ωlimit sets with the Hausdorff metric. Real Anal. Exchange 24(2) (1998/99), [8] A. Blokh, A. Bruckner, P. Humke and J. Smítal. The space of ωlimit sets of a continuous map of the interval. Trans. Amer. Math. Soc. 348(4) (1996), [9] A. M. Blokh. The spectral decomposition for onedimensional maps. Dynamics Reported (Dynamics Reported, Expositions in Dynamical Systems (New Series), 4). Springer, Berlin, 1995, pp [1] C. Bonatti, L. Diaz and M. Viana. Dynamics Beyond Hyperbolicity. Springer, Berlin, 25. [11] R. Bowen. Markov partitions for axiom a diffeomorphisms. Amer. J. Math. 92 (197), [12] C. Conley. Isolated Invariant Sets and the Morse Index. American Mathematical Society, Providence, RI, [13] L. Díaz and J. Rocha. Heterodimensional cycles, partial hyperbolicity and limit dynamics. Fund. Math. 174(2) (22), [14] V. Horita, N. Muniz and P. R. Sabini. Nonperiodic bifurcations of onedimensional maps. Ergod. Th. & Dynam. Sys. 27(2) (27), [15] V. J. López. Period doubling is the boundary of chaos and of order in the C 1 topology of interval maps. Nonlinearity 15(3) (22), [16] R. Mañé. Hyperbolicity, sinks and measure in onedimensional dynamics. Comm. Math. Phys. 1(4) (1985), [17] F. Marotto. Snapback repellers imply chaos in R n. J. Math. Anal. Appl. 63 (1978), [18] M. Martens, W. de Melo and S. van Strien. JuliaFatouSullivan theory for real onedimensional dynamics. Acta Math. 168(3 4) (1992), [19] S. Newhouse and J. Palis. Cycles and bifurcation theory. Astérisque 31(44 14) (1976). [2] J. Palis and F. Takens. Hyperbolicity and the creation of homoclinic orbits. Ann. of Math. (2) 125 (1987), [21] J. Palis and F. Takens. Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, [22] S. E. Patterson. stable limit set explosions. Trans. Amer. Math. Soc. 294(2) (1986), [23] C. Robert, K. T. Alligood, E. Ott and J. A. Yorke. Explosions of chaotic sets. Phys. D 144(1 2) (2), [24] P. R. Sabini. Nonperiodic bifurcations at the boundary of hyperbolic systems. PhD Thesis, IMPA, 21. [25] E. Sander. Homoclinic tangles for noninvertible maps. Nonlinear Anal. 41(1 2) (2),
A classification of explosions in dimension one
arxiv:math/0703176v2 [math.ds] 6 Mar 2007 A classification of explosions in dimension one E. Sander  J.A. Yorke Preprint July 30, 2018 Abstract A discontinuous change inthesize of anattractor is themost
More informationHYPERBOLIC SETS WITH NONEMPTY INTERIOR
HYPERBOLIC SETS WITH NONEMPTY INTERIOR TODD FISHER, UNIVERSITY OF MARYLAND Abstract. In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic
More informationEssential hyperbolicity versus homoclinic bifurcations. Global dynamics beyond uniform hyperbolicity, Beijing 2009 Sylvain Crovisier  Enrique Pujals
Essential hyperbolicity versus homoclinic bifurcations Global dynamics beyond uniform hyperbolicity, Beijing 2009 Sylvain Crovisier  Enrique Pujals Generic dynamics Consider: M: compact boundaryless manifold,
More informationC 1 DENSITY OF AXIOM A FOR 1D DYNAMICS
C 1 DENSITY OF AXIOM A FOR 1D DYNAMICS DAVID DIICA Abstract. We outline a proof of the C 1 unimodal maps of the interval. density of Axiom A systems among the set of 1. Introduction The amazing theory
More informationHyperbolic Dynamics. Todd Fisher. Department of Mathematics University of Maryland, College Park. Hyperbolic Dynamics p.
Hyperbolic Dynamics p. 1/36 Hyperbolic Dynamics Todd Fisher tfisher@math.umd.edu Department of Mathematics University of Maryland, College Park Hyperbolic Dynamics p. 2/36 What is a dynamical system? Phase
More informationWHAT IS A CHAOTIC ATTRACTOR?
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties
More informationThe Structure of Hyperbolic Sets
The Structure of Hyperbolic Sets p. 1/35 The Structure of Hyperbolic Sets Todd Fisher tfisher@math.umd.edu Department of Mathematics University of Maryland, College Park The Structure of Hyperbolic Sets
More informationCHAOTIC UNIMODAL AND BIMODAL MAPS
CHAOTIC UNIMODAL AND BIMODAL MAPS FRED SHULTZ Abstract. We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal
More informationTHE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM
Bull. Aust. Math. Soc. 88 (2013), 267 279 doi:10.1017/s0004972713000348 THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM MICHAEL F. BARNSLEY and ANDREW VINCE (Received 15 August 2012; accepted 21 February
More informationDYNAMICAL SYSTEMS. I Clark: Robinson. Stability, Symbolic Dynamics, and Chaos. CRC Press Boca Raton Ann Arbor London Tokyo
DYNAMICAL SYSTEMS Stability, Symbolic Dynamics, and Chaos I Clark: Robinson CRC Press Boca Raton Ann Arbor London Tokyo Contents Chapter I. Introduction 1 1.1 Population Growth Models, One Population 2
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005  Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationDYNAMICAL SYSTEMS PROBLEMS. asgor/ (1) Which of the following maps are topologically transitive (minimal,
DYNAMICAL SYSTEMS PROBLEMS http://www.math.uci.edu/ asgor/ (1) Which of the following maps are topologically transitive (minimal, topologically mixing)? identity map on a circle; irrational rotation of
More informationSubalpine Rhapsody in Dynamics CONTENTS
RENDICONTI DEL SEMINARIO MATEMATICO Università e Politecnico di Torino Subalpine Rhapsody in Dynamics CONTENTS K.T. Alligood  E. Sander  J.A. Yorke, Explosions in dimensions one through three.....................................
More informationSHADOWING AND INTERNAL CHAIN TRANSITIVITY
SHADOWING AND INTERNAL CHAIN TRANSITIVITY JONATHAN MEDDAUGH AND BRIAN E. RAINES Abstract. The main result of this paper is that a map f : X X which has shadowing and for which the space of ωlimits sets
More informationPersonal notes on renormalization
Personal notes on renormalization Pau Rabassa Sans April 17, 2009 1 Dynamics of the logistic map This will be a fast (and selective) review of the dynamics of the logistic map. Let us consider the logistic
More informationRenormalization for Lorenz maps
Renormalization for Lorenz maps Denis Gaidashev, Matematiska Institutionen, Uppsala Universitet Tieste, June 5, 2012 D. Gaidashev, Uppsala Universitet () Renormalization for Lorenz maps Tieste, June 5,
More informationTRIVIAL CENTRALIZERS FOR AXIOM A DIFFEOMORPHISMS
TRIVIAL CENTRALIZERS FOR AXIOM A DIFFEOMORPHISMS TODD FISHER Abstract. We show there is a residual set of nonanosov C Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer.
More informationThe real analytic FeigenbaumCoulletTresser attractor in the disk.
The real analytic FeigenbaumCoulletTresser attractor in the disk. Eleonora Catsigeras, Marcelo Cerminara and Heber Enrich Instituto de Matemática (IMERL), Facultad de Ingenería Universidad de la República.
More informationTOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS
Chung, Y.M. Osaka J. Math. 38 (200), 2 TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS YONG MOO CHUNG (Received February 9, 998) Let Á be a compact interval of the real line. For a continuous
More informationarxiv: v1 [math.ds] 16 Nov 2010
Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms arxiv:1011.3836v1 [math.ds] 16 Nov 2010 Sylvain Crovisier Enrique R. Pujals October 30, 2018 Abstract
More informationPERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA
PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS A. M. Blokh Department of Mathematics, Wesleyan University Middletown, CT 064590128, USA August 1991, revised May 1992 Abstract. Let X be a compact tree,
More informationFinding numerically Newhouse sinks near a homoclinic tangency and investigation of their chaotic transients. Takayuki Yamaguchi
Hokkaido Mathematical Journal Vol. 44 (2015) p. 277 312 Finding numerically Newhouse sinks near a homoclinic tangency and investigation of their chaotic transients Takayuki Yamaguchi (Received March 13,
More information1 Introduction Definitons Markov... 2
Compact course notes Dynamic systems Fall 2011 Professor: Y. Kudryashov transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Introduction 2 1.1 Definitons...............................................
More informationRobustly transitive diffeomorphisms
Robustly transitive diffeomorphisms Todd Fisher tfisher@math.byu.edu Department of Mathematics, Brigham Young University Summer School, Chengdu, China 2009 Dynamical systems The setting for a dynamical
More informationORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY
ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY AND ωlimit SETS CHRIS GOOD AND JONATHAN MEDDAUGH Abstract. Let f : X X be a continuous map on a compact metric space, let ω f be the collection of ωlimit
More informationc/o Dipartimento di Matematica dell Università di Torino
Aims The Seminario Matematico is a society of members of the Mathematicsrelated Departments in the University and Polytechnic of Turin. Its scope is to promote study and research in all fields of Mathematics
More informationPersistent Chaos in HighDimensional Neural Networks
Persistent Chaos in HighDimensional Neural Networks D. J. Albers with J. C. Sprott and James P. Crutchfield February 20, 2005 1 Outline: Introduction and motivation Mathematical versus computational dynamics
More informationSymbolic dynamics and chaos in plane Couette flow
Dynamics of PDE, Vol.14, No.1, 7985, 2017 Symbolic dynamics and chaos in plane Couette flow Y. Charles Li Communicated by Y. Charles Li, received December 25, 2016. Abstract. According to a recent theory
More informationTECHNIQUES FOR ESTABLISHING DOMINATED SPLITTINGS
TECHNIQUES FOR ESTABLISHING DOMINATED SPLITTINGS ANDY HAMMERLINDL ABSTRACT. We give theorems which establish the existence of a dominated splitting and further properties, such as partial hyperbolicity.
More informationPeriodic Sinks and Observable Chaos
Periodic Sinks and Observable Chaos Systems of Study: Let M = S 1 R. T a,b,l : M M is a threeparameter family of maps defined by where θ S 1, r R. θ 1 = a+θ +Lsin2πθ +r r 1 = br +blsin2πθ Outline of Contents:
More informationMARKOV PARTITIONS FOR HYPERBOLIC SETS
MARKOV PARTITIONS FOR HYPERBOLIC SETS TODD FISHER, HIMAL RATHNAKUMARA Abstract. We show that if f is a diffeomorphism of a manifold to itself, Λ is a mixing (or transitive) hyperbolic set, and V is a neighborhood
More informationThe Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania
The Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania AMS Subject Classification: Primary: 37F10 Secondary: 51N05, 54D70 Key Words: Julia Set, Complex Dynamics,
More informationThe spectral decomposition for onedimensional maps
The spectral decomposition for onedimensional maps Alexander M.Blokh 1 Department of Mathematics, Wesleyan University Middletown, CT 064576035, USA Abstract. We construct the spectral decomposition of
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationTHE CHAIN RECURRENT SET FOR MAPS OF THE CIRCLE
proceedings of the american mathematical society Volume 92, Number 4, December 1984 THE CHAIN RECURRENT SET FOR MAPS OF THE CIRCLE LOUIS BLOCK AND JOHN E. FRANKE ABSTRACT. For a continuous map of the circle
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA Email: topolog@auburn.edu ISSN: 01464124
More informationSINGULAR HYPERBOLIC SYSTEMS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 11, Pages 3393 3401 S 00029939(99)049369 Article electronically published on May 4, 1999 SINGULAR HYPERBOLIC SYSTEMS C. A. MORALES,
More informationChaos in the Dynamics of the Family of Mappings f c (x) = x 2 x + c
IOSR Journal of Mathematics (IOSRJM) eissn: 78578, pissn: 319765X. Volume 10, Issue 4 Ver. IV (JulAug. 014), PP 108116 Chaos in the Dynamics of the Family of Mappings f c (x) = x x + c Mr. Kulkarni
More informationSOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS. 1. Introduction
SOME COUNTEREXAMPLES IN DYNAMICS OF RATIONAL SEMIGROUPS RICH STANKEWITZ, TOSHIYUKI SUGAWA, AND HIROKI SUMI Abstract. We give an example of two rational functions with nonequal Julia sets that generate
More informationDynamics of Tangent. Robert L. Devaney Department of Mathematics Boston University Boston, Mass Linda Keen
Dynamics of Tangent Robert L. Devaney Department of Mathematics Boston University Boston, Mass. 02215 Linda Keen Department of Mathematics Herbert H. Lehman College, CUNY Bronx, N.Y. 10468 Abstract We
More informationJULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 JULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS BY ROBERT L. DEVANEY ABSTRACT. We describe some of the
More informationHomoclinic tangency and variation of entropy
CADERNOS DE MATEMÁTICA 10, 133 143 May (2009) ARTIGO NÚMERO SMA# 313 Homoclinic tangency and variation of entropy M. Bronzi * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação,
More informationMathematische Annalen
Math. Ann. 334, 457 464 (2006) Mathematische Annalen DOI: 10.1007/s0020800507432 The Julia Set of Hénon Maps John Erik Fornæss Received:6 July 2005 / Published online: 9 January 2006 SpringerVerlag
More informationOn dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I
IOP PUBLISHING Nonlinearity 2 (28) 923 972 NONLINEARITY doi:.88/95775/2/5/3 On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I S V Gonchenko, L P Shilnikov and D V Turaev
More informationZin ARAI Hokkaido University
荒井迅 Zin ARAI Hokkaido University Yutaka Ishii (Kyushu University) Background and Main Results The Hénon Map The Hénon family on R 2 : f a,b : R 2 3 (x, y) 7! (x 2 a by, x) 2 R 2, where (a, b) 2 R R. 1
More informationHAMILTONIAN ELLIPTIC DYNAMICS ON SYMPLECTIC 4MANIFOLDS
HAMILTONIAN ELLIPTIC DYNAMICS ON SYMPLECTIC 4MANIFOLDS MÁRIO BESSA AND JOÃO LOPES DIAS Abstract. We consider C 2 Hamiltonian functions on compact 4dimensional symplectic manifolds to study elliptic dynamics
More informationThe Existence of Chaos in the Lorenz System
The Existence of Chaos in the Lorenz System Sheldon E. Newhouse Mathematics Department Michigan State University E. Lansing, MI 48864 joint with M. Berz, K. Makino, A. Wittig Physics, MSU Y. Zou, Math,
More informationAbundance of stable ergodicity
Abundance of stable ergodicity Christian Bonatti, Carlos Matheus, Marcelo Viana, Amie Wilkinson October 5, 2004 Abstract We consider the set PH ω (M) of volume preserving partially hyperbolic diffeomorphisms
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integrodifferential equations
More informationIntroduction to Continuous Dynamical Systems
Lecture Notes on Introduction to Continuous Dynamical Systems Fall, 2012 Lee, Keonhee Department of Mathematics Chungnam National Univeristy  1  Chap 0. Introduction What is a dynamical system? A dynamical
More informationFOURTH ORDER CONSERVATIVE TWIST SYSTEMS: SIMPLE CLOSED CHARACTERISTICS
FOURTH ORDER CONSERVATIVE TWIST SYSTEMS: SIMPLE CLOSED CHARACTERISTICS J.B. VAN DEN BERG AND R.C.A.M. VANDERVORST ABSTRACT. On the energy manifolds of fourth order conservative systems closed characteristics
More informationEntropy production for a class of inverse SRB measures
Entropy production for a class of inverse SRB measures Eugen Mihailescu and Mariusz Urbański Keywords: Inverse SRB measures, folded repellers, Anosov endomorphisms, entropy production. Abstract We study
More informationDynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and
Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly
More informationContinuumWise Expansive and Dominated Splitting
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 23, 11491154 HIKARI Ltd, www.mhikari.com ContinuumWise Expansive and Dominated Splitting Manseob Lee Department of Mathematics Mokwon University Daejeon,
More informationMathematical Foundations of Neuroscience  Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience  Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationRemoving the Noise from Chaos Plus Noise
Removing the Noise from Chaos Plus Noise Steven P. Lalley Department of Statistics University of Chicago November 5, 2 Abstract The problem of extracting a signal x n generated by a dynamical system from
More informationCOARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS
COARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS STEVEN FRANKEL Abstract. We prove Calegari s conjecture that every quasigeodesic flow on a closed hyperbolic 3manifold has closed orbits.
More informationTOPOLOGICAL ENTROPY AND TOPOLOGICAL STRUCTURES OF CONTINUA
TOPOLOGICAL ENTROPY AND TOPOLOGICAL STRUCTURES OF CONTINUA HISAO KATO, INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA 1. Introduction During the last thirty years or so, many interesting connections between
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationAPPLICATIONS OF ALMOST ONETOONE MAPS
APPLICATIONS OF ALMOST ONETOONE MAPS ALEXANDER BLOKH, LEX OVERSTEEGEN, AND E. D. TYMCHATYN Abstract. A continuous map f : X Y of topological spaces X, Y is said to be almost 1to1 if the set of the
More informationResearch Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane
Hindawi Publishing Corporation Advances in Difference Equations Volume 009 Article ID 1380 30 pages doi:101155/009/1380 Research Article Global Dynamics of a Competitive System of Rational Difference Equations
More informationTowards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University
Towards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University Dynamical systems with multiple time scales arise naturally in many domains. Models of neural systems
More informationPATH CONNECTEDNESS AND ENTROPY DENSITY OF THE SPACE OF HYPERBOLIC ERGODIC MEASURES
PATH CONNECTEDNESS AND ENTROPY DENSITY OF THE SPACE OF HYPERBOLIC ERGODIC MEASURES ANTON GORODETSKI AND YAKOV PESIN Abstract. We show that the space of hyperbolic ergodic measures of a given index supported
More informationORIENTATIONREVERSING MORSESMALE DIFFEOMORPHISMS ON THE TORUS
TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 264, Number 1, March 1981 ORIENTATIONREVERSING MORSESMALE DIFFEOMORPHISMS ON THE TORUS BY STEVE BATTERSON1 Abstract. For orientationreversing
More informationThe Homoclinic Tangle and Smale s Horseshoe Map
The Homoclinic Tangle and Smale s Horseshoe Map Matt I. Introduction The homoclinic tangle is an interesting property of the manifolds of certain maps. In order to give more formal meaning to some intuitive
More informationFAMILIES OF TRANSITIVE MAPS ON R WITH HORIZONTAL ASYMPTOTES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 59, No. 2, 2018, Pages 375 387 Published online: May 28, 2018 FAMILIES OF TRANSITIVE MAPS ON R WITH HORIZONTAL ASYMPTOTES BLADISMIR LEAL, GUELVIS MATA, AND
More informationPHY411 Lecture notes Part 5
PHY411 Lecture notes Part 5 Alice Quillen January 27, 2016 Contents 0.1 Introduction.................................... 1 1 Symbolic Dynamics 2 1.1 The Shift map.................................. 3 1.2
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationarxiv: v1 [math.ds] 6 Apr 2011
STABILIZATION OF HETERODIMENSIONAL CYCLES C. BONATTI, L. J. DÍAZ, AND S. KIRIKI arxiv:1104.0980v1 [math.ds] 6 Apr 2011 Abstract. We consider diffeomorphisms f with heteroclinic cycles associated to saddles
More informationCharacterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddlenode equilibrium point of type 0
Anais do CNMAC v.2 ISSN 198482X Characterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddlenode equilibrium point of type Fabíolo M. Amaral Departamento
More informationOn the dynamics of dominated splitting
Annals of Mathematics, 169 (2009), 675 740 On the dynamics of dominated splitting By Enrique R. Pujals and Martín Sambarino* To Jose Luis Massera, in memorian Abstract Let f : M M be a C 2 diffeomorphism
More informationZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS
ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS MICHAEL C. SULLIVAN Abstract. The Zeeman number is an invariant of flow equivalence of suspensions of shifts of finite type. We prove that the Zeeman number
More informationarxiv: v2 [math.ca] 4 Jun 2017
EXCURSIONS ON CANTORLIKE SETS ROBERT DIMARTINO AND WILFREDO O. URBINA arxiv:4.70v [math.ca] 4 Jun 07 ABSTRACT. The ternary Cantor set C, constructed by George Cantor in 883, is probably the best known
More informationThe centralizer of a C 1 generic diffeomorphism is trivial
The centralizer of a C 1 generic diffeomorphism is trivial Christian Bonatti, Sylvain Crovisier and Amie Wilkinson April 16, 2008 Abstract Answering a question of Smale, we prove that the space of C 1
More informationAbundance of stable ergodicity
Abundance of stable ergodicity Christian Bonatti, Carlos atheus, arcelo Viana, Amie Wilkinson December 7, 2002 Abstract We consider the set PH ω () of volume preserving partially hyperbolic diffeomorphisms
More informationTHE CLASSIFICATION OF TILING SPACE FLOWS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 THE CLASSIFICATION OF TILING SPACE FLOWS by Alex Clark Abstract. We consider the conjugacy of the natural flows on onedimensional tiling
More informationCorrelation dimension for selfsimilar Cantor sets with overlaps
F U N D A M E N T A MATHEMATICAE 155 (1998) Correlation dimension for selfsimilar Cantor sets with overlaps by Károly S i m o n (Miskolc) and Boris S o l o m y a k (Seattle, Wash.) Abstract. We consider
More informationUNCOUNTABLE ωlimit SETS WITH ISOLATED POINTS
UNCOUNTABLE ωlimit SETS WITH ISOLATED POINTS CHRIS GOOD, BRIAN RAINES, AND ROLF SUABEDISSEN Abstract. We give two examples of tent maps with uncountable (as it happens, postcritical) ωlimit sets, which
More informationOn surfaceknots with triple point number at most three
On surfaceknots with triple point number at most three A. Al Kharusi and T. Yashiro Abstract arxiv:1711.04838v1 [math.at] 13 Nov 2017 In this paper, we show that if a diagram of a surfaceknot F has at
More informationDynamical Systems 2, MA 761
Dynamical Systems 2, MA 761 Topological Dynamics This material is based upon work supported by the National Science Foundation under Grant No. 9970363 1 Periodic Points 1 The main objects studied in the
More information(Non)Existence of periodic orbits in dynamical systems
(Non)Existence of periodic orbits in dynamical systems Konstantin Athanassopoulos Department of Mathematics and Applied Mathematics University of Crete June 3, 2014 onstantin Athanassopoulos (Univ. of
More informationExample of a Blue Sky Catastrophe
PUB:[SXG.TEMP]TRANS2913EL.PS 16OCT2001 11:08:53.21 SXG Page: 99 (1) Amer. Math. Soc. Transl. (2) Vol. 200, 2000 Example of a Blue Sky Catastrophe Nikolaĭ Gavrilov and Andrey Shilnikov To the memory of
More information(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA Email: topolog@auburn.edu ISSN: 01464124
More informationarxiv: v1 [math.ds] 31 Jul 2018
arxiv:1807.11801v1 [math.ds] 31 Jul 2018 On the interior of projections of planar selfsimilar sets YUKI TAKAHASHI Abstract. We consider projections of planar selfsimilar sets, and show that one can create
More informationSingular Perturbations in the McMullen Domain
Singular Perturbations in the McMullen Domain Robert L. Devaney Sebastian M. Marotta Department of Mathematics Boston University January 5, 2008 Abstract In this paper we study the dynamics of the family
More informationarxiv: v2 [math.ds] 3 Jan 2019
SHADOWING, ASYMPTOTIC SHADOWING AND SLIMIT SHADOWING CHRIS GOOD, PIOTR OPROCHA, AND MATE PULJIZ arxiv:1701.05383v2 [math.ds] 3 Jan 2019 Abstract. We study three notions of shadowing: classical shadowing,
More informationThe Conley Index and Rigorous Numerics of Attracting Periodic Orbits
The Conley Index and Rigorous Numerics of Attracting Periodic Orbits Marian Mrozek Pawe l Pilarczyk Conference on Variational and Topological Methods in the Study of Nonlinear Phenomena (Pisa, 2000) 1
More informationEntropy of C 1 diffeomorphisms without dominated splitting
Entropy of C 1 diffeomorphisms without dominated splitting Jérôme Buzzi (CNRS & Université ParisSud) joint with S. CROVISIER and T. FISHER June 15, 2017 Beyond Uniform Hyperbolicity  Provo, UT Outline
More informationTHE SET OF RECURRENT POINTS OF A CONTINUOUS SELFMAP ON AN INTERVAL AND STRONG CHAOS
J. Appl. Math. & Computing Vol. 4(2004), No.  2, pp. 277288 THE SET OF RECURRENT POINTS OF A CONTINUOUS SELFMAP ON AN INTERVAL AND STRONG CHAOS LIDONG WANG, GONGFU LIAO, ZHENYAN CHU AND XIAODONG DUAN
More informationSufficient conditions for a period incrementing big bang bifurcation in onedimensional maps.
Sufficient conditions for a period incrementing big bang bifurcation in onedimensional maps. V. Avrutin, A. Granados and M. Schanz Abstract Typically, big bang bifurcation occur for one (or higher)dimensional
More informationCHARACTERIZATION OF SADDLENODE EQUILIBRIUM POINTS ON THE STABILITY BOUNDARY OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEM
CHARACTERIZATION OF SADDLENODE EQUILIBRIUM POINTS ON THE STABILITY BOUNDARY OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEM Fabíolo Moraes Amaral, Josaphat Ricardo Ribeiro Gouveia Júnior, Luís Fernando Costa
More informationDynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
More informationTHE SEMICONTINUITY LEMMA
THE SEMICONTINUITY LEMMA BRUNO SANTIAGO Abstract. In this note we present a proof of the wellknow semicontinuity lemma in the separable case. 1. Introduction There exists two basic topological results,
More informationInvariant manifolds of the Bonhoeffervan der Pol oscillator
Invariant manifolds of the Bonhoeffervan der Pol oscillator R. Benítez 1, V. J. Bolós 2 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda. Virgen del Puerto 2. 10600,
More informationEquilibrium States for Partially Hyperbolic Horseshoes
Equilibrium States for Partially Hyperbolic Horseshoes R. Leplaideur, K. Oliveira, and I. Rios August 25, 2009 Abstract We study ergodic properties of invariant measures for the partially hyperbolic horseshoes,
More informationCherry flows with non trivial attractors
Cherry flows with non trivial attractors Liviana Palmisano Institute of Mathematics of PAN Warsaw 00950, Poland January 17, 2018 arxiv:1509.05994v2 [math.ds] 4 Sep 2016 Abstract We provide an example
More informationProblem: A class of dynamical systems characterized by a fast divergence of the orbits. A paradigmatic example: the Arnold cat.
À È Ê ÇÄÁ Ë ËÌ ÅË Problem: A class of dynamical systems characterized by a fast divergence of the orbits A paradigmatic example: the Arnold cat. The closure of a homoclinic orbit. The shadowing lemma.
More information